4.4.1 Volume conservation

Consider a shallow layer of fluid with a free surface as shown in Figure 4.1. For definiteness, such a fluid layer can be considered the fluid which occupies the surface layer of the ocean model. The position $z=\eta(\lambda,\phi,t)$, which could be negative, is the vertical deviation from a resting ocean state z=0. The position z=z₁ < 0 is the fixed position of the bottom of the layer. The volume of an infinitesimally thin column of fluid extending over the finite vertical extent of this layer is given by

$$\delta V = h \, \delta A,$$ (4.35)

where

$$h = -z_{1} + \eta$$ (4.36)

is the height of the fluid column, and $\delta A$ is its infinitesimal horizontal cross-sectional area. Volume conservation implies that the change of the box volume with time equals the sum of all volume fluxes through the box surface,

$$\partial_{t}(\delta V) \left(q_{w} + w_{1} -\nabla_{h} \cdot\int_{z_1}^\eta dz \, {\bf u}_{h} \right) \, \delta A.$$ = (4.37)

The convergence of the horizontal flux stems from the infinitesimal horizontal extension of the box. $w_{1} \, \delta A$ is the volume per unit time crossing through the bottom of the layer, where w₁ = w(z=z₁) is positive for water moving upward into the surface layer. $q_{w} \, \delta A$ is the volume per unit time of ocean water appearing in the surface box. In the spirit of a volume conserving model, it is equivalent to the volume of fresh water per unit time crossing the free surface, with q_w>0 indicating water entering the ocean. The accuracy of this equivalence is determined by the deviation of the ratio of the fresh water density $\rho_f$ and the ocean density from unity

$$\frac{\rho_f}{\rho(z=\eta)} -1.$$ (4.38)

For the most applications this deviation should be smaller than the accuracy of the fresh water flux data.

The identity

$$\partial_{t}(\delta V) \delta A \, \partial_{t} h$$ = (4.39)

leads to the balance for the layer thickness

 

$$\partial_{t}h + \nabla_{h} \cdot \int_{z_1}^\eta {\bf u}_{h}$$ = q_w + w₁. (4.40)

With a uniform velocity in the surface layer, this result takes the more familiar form

 

$$\partial_{t}h + \nabla_{h} \cdot (h \, {\bf u}_{h})$$ = q_w + w₁. (4.41)